Proving Theorem i.5 with Calvo Apostol

cnaeger · Oct 26, 2007

Hello,

I am working my way through Cal vo1 by Apostol and working the problems and was wondering if someone could assist.

The question is on pg 19 to prove theorems i.5-i.15 using axioms and theorems i.1-i.4.

Theorem i.5 states a(b-c) = ab - bc.

I started the proof but am stuck at at the point indicated below.

a(b-c) = a(b + (-c)) = ab + a(-c)

being able to represent a(-c) as -ac is not illustrated until theorem i.12.

I tried representing a(-c) as -(-a)(-c) but that did not go anywhere either due to (-a)(-c) not being illustrated until i.12 either.

Can anyone help me along with this one. This is not a homework assignment as I am a BS in EE with a minor in math. Just working my way through this book to enhance my knowledge.

Thanks,
Chad

Gib Z · Oct 26, 2007

I personally don't have Apostol, though others on this forum most likely do. Either wait for one of them to post, or please post here stating which Theorems and Axioms you are given, ie what you are already allowed to assume.

neutrino · Oct 27, 2007

a(b-c) = a(b + (-c)) = ab + a(-c)

ab+a(-c) + ac = ?

bmm9 · Oct 27, 2007

Prove that ac + a(-c) = 0... then a(-c) = -ac.. I suppose you have to prove that additive inverses are unique but you can use the cancellation law for this. Don't stress over this part too much; most proofs aren't like this. If you did most of the set proofs you should be good - generally that is the stuff that is unfamiliar compared to field axioms.

cnaeger · Oct 27, 2007

In response to Gib Z the Axioms and theorems available are:

A.1 x + y = y + x, xy = yx (Comm Laws)
A.2 x + (y + z) = (x + y) + z, x(yz) = (xy)z (Assoc Laws)
A.3 x(y + z) = xy + xz (Dist Law)
A.4 x + 0 = x, 1*x = x (Existence of Identity elements)
A.5 For every real # x there is a real # y such that x + y =0 (Existence of negatives)
A.6 For every real # x not equal to 0 there is a number y such that xy=1 (Reciprocals)

Th.1 If a + b = a + c then b=c (Cancellation law for addition)
Th.2 Given a and b there is exactly one x such that a + x = b. This is denoted by b - a. In particular 0 - a is simply written -a and called the negative of a.
Th.3 b - a = b + (-a)
Th.4 -(-a) = a

Thanks,
Chad

bmm9 · Oct 27, 2007

So far, you have:
a(b-c) = a(b + (-c)) = ab + a(-c)

Now you need to show that a(-c) = -ac... what property does -ac have that you suspect a(-c) has? they're both additive inverses of ac, so..
ac - ac = a(c - c) by a.3
and ac + a(-c) = a(c + (-c)) = a(c - c),
so ac - ac = ac + a(-c), and by thm.1, -ac = a(-c),
now just plug this back into what you had (or you could work from that form the entire time, just leaving "ab" in).

Proving Theorem i.5 with Calvo Apostol

1. What is Theorem i.5 in Calvo Apostol?
Theorem i.5 states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This is commonly known as the Pythagorean Theorem.

2. How do you prove Theorem i.5 with Calvo Apostol?
The proof of Theorem i.5 involves using the concepts of similar triangles, the Pythagorean Theorem, and the properties of right triangles. It is a rigorous mathematical proof that follows the logical steps outlined in Calvo Apostol's book.

3. Why is Theorem i.5 important?
Theorem i.5 is important because it is a fundamental principle in geometry and has numerous applications in mathematics, physics, and other fields. It is also the basis for many other theorems and geometric proofs.

4. Can Theorem i.5 be proven using other methods?
Yes, Theorem i.5 can be proven using various methods such as algebraic proofs, geometric proofs, and even visual proofs. However, the proof in Calvo Apostol's book is a classic and well-respected method.

Similar threads

Hot Threads

Recent Insights

Proving Theorem i.5 with Calvo Apostol

1. What is Theorem i.5 in Calvo Apostol? Theorem i.5 states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This is commonly known as the Pythagorean Theorem.

2. How do you prove Theorem i.5 with Calvo Apostol? The proof of Theorem i.5 involves using the concepts of similar triangles, the Pythagorean Theorem, and the properties of right triangles. It is a rigorous mathematical proof that follows the logical steps outlined in Calvo Apostol's book.

3. Why is Theorem i.5 important? Theorem i.5 is important because it is a fundamental principle in geometry and has numerous applications in mathematics, physics, and other fields. It is also the basis for many other theorems and geometric proofs.

4. Can Theorem i.5 be proven using other methods? Yes, Theorem i.5 can be proven using various methods such as algebraic proofs, geometric proofs, and even visual proofs. However, the proof in Calvo Apostol's book is a classic and well-respected method.

Similar threads

Hot Threads

Recent Insights

1. What is Theorem i.5 in Calvo Apostol?
Theorem i.5 states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This is commonly known as the Pythagorean Theorem.

2. How do you prove Theorem i.5 with Calvo Apostol?
The proof of Theorem i.5 involves using the concepts of similar triangles, the Pythagorean Theorem, and the properties of right triangles. It is a rigorous mathematical proof that follows the logical steps outlined in Calvo Apostol's book.

3. Why is Theorem i.5 important?
Theorem i.5 is important because it is a fundamental principle in geometry and has numerous applications in mathematics, physics, and other fields. It is also the basis for many other theorems and geometric proofs.

4. Can Theorem i.5 be proven using other methods?
Yes, Theorem i.5 can be proven using various methods such as algebraic proofs, geometric proofs, and even visual proofs. However, the proof in Calvo Apostol's book is a classic and well-respected method.